dg.differential geometry - Pairing used in Lefschetz duality

I am thinking about the precise formulation of the Lefschetz duality for the relative cohomology. If I understand this Wikipedia article correctly, there is an isomorphism between $H^k(M, partial M)$ and $H_{n-k}(M)$ and hence (I suppose) a non-degenerate pairing $H^k(M, partial M) times H^{n-k}(M) rightarrow mathbb{R}$. However, I have trouble visualizing this pairing. Let $[(alpha, theta)] in H^k(M, partial M)$ and $[beta] in H^{n - k}(M)$, is it then true that

$$left< [(alpha, theta)], [beta] right> = int_M alpha wedge beta + int_{partial M}theta wedge beta_{|partial M}$$
or am I missing something? If unrelated to Lefschetz duality, does this pairing ever appear in topology?

I can understand how to define a pairing on the homology by counting intersections, but I really don't see how this works for cohomology. Also, a reference on Lefschetz cohomology or just analysis/topology on manifolds with boundary would be greatly appreciated!

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